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Which of the following best models the data in the table?

Type of Function

The table presents values of x and y, showing how y increases as x increases
x y
2
8.5

4
46

6
107.5

8
193

(1 point)
Responses

linear
linear

quadratic
quadratic

cubic
cubic

exponential

1 answer

To determine which type of function best models the data provided in the table, we can analyze the values of \(y\) in relation to \(x\).

Given:

  • \(x = 2\), \(y = 8.5\)
  • \(x = 4\), \(y = 46\)
  • \(x = 6\), \(y = 107.5\)
  • \(x = 8\), \(y = 193\)

Let's look at the differences between consecutive \(y\) values:

  1. From \(x = 2\) to \(x = 4\): \(46 - 8.5 = 37.5\)

  2. From \(x = 4\) to \(x = 6\): \(107.5 - 46 = 61.5\)

  3. From \(x = 6\) to \(x = 8\): \(193 - 107.5 = 85.5\)

Now let's calculate the second differences (differences of the first differences):

  • First differences: \(37.5\), \(61.5\), \(85.5\)
  1. From \(37.5\) to \(61.5\): \(61.5 - 37.5 = 24\)
  2. From \(61.5\) to \(85.5\): \(85.5 - 61.5 = 24\)

The second differences are constant (both equal to 24), which suggests that the relationship between \(x\) and \(y\) is quadratic.

Thus, the best model for the data in the table is:

quadratic